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Theorem infdifsn 8771
Description: Removing a singleton from an infinite set does not change the cardinality of the set. (Contributed by Mario Carneiro, 30-Apr-2015.) (Revised by Mario Carneiro, 16-May-2015.)
Assertion
Ref Expression
infdifsn (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)

Proof of Theorem infdifsn
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 brdomi 8173 . . . 4 (ω ≼ 𝐴 → ∃𝑓 𝑓:ω–1-1𝐴)
21adantr 472 . . 3 ((ω ≼ 𝐴𝐵𝐴) → ∃𝑓 𝑓:ω–1-1𝐴)
3 reldom 8168 . . . . . . 7 Rel ≼
43brrelex2i 5331 . . . . . 6 (ω ≼ 𝐴𝐴 ∈ V)
54ad2antrr 717 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐴 ∈ V)
6 simplr 785 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐵𝐴)
7 f1f 6285 . . . . . . 7 (𝑓:ω–1-1𝐴𝑓:ω⟶𝐴)
87adantl 473 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω⟶𝐴)
9 peano1 7285 . . . . . 6 ∅ ∈ ω
10 ffvelrn 6549 . . . . . 6 ((𝑓:ω⟶𝐴 ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ 𝐴)
118, 9, 10sylancl 580 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓‘∅) ∈ 𝐴)
12 difsnen 8251 . . . . 5 ((𝐴 ∈ V ∧ 𝐵𝐴 ∧ (𝑓‘∅) ∈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
135, 6, 11, 12syl3anc 1490 . . . 4 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}))
14 vex 3353 . . . . . . . . . 10 𝑓 ∈ V
15 f1f1orn 6333 . . . . . . . . . . 11 (𝑓:ω–1-1𝐴𝑓:ω–1-1-onto→ran 𝑓)
1615adantl 473 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1-onto→ran 𝑓)
17 f1oen3g 8178 . . . . . . . . . 10 ((𝑓 ∈ V ∧ 𝑓:ω–1-1-onto→ran 𝑓) → ω ≈ ran 𝑓)
1814, 16, 17sylancr 581 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ ran 𝑓)
1918ensymd 8213 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓 ≈ ω)
203brrelex1i 5330 . . . . . . . . . . 11 (ω ≼ 𝐴 → ω ∈ V)
2120ad2antrr 717 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ∈ V)
22 limom 7280 . . . . . . . . . . 11 Lim ω
2322limenpsi 8344 . . . . . . . . . 10 (ω ∈ V → ω ≈ (ω ∖ {∅}))
2421, 23syl 17 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ (ω ∖ {∅}))
2514resex 5622 . . . . . . . . . . 11 (𝑓 ↾ (ω ∖ {∅})) ∈ V
26 simpr 477 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓:ω–1-1𝐴)
27 difss 3901 . . . . . . . . . . . 12 (ω ∖ {∅}) ⊆ ω
28 f1ores 6336 . . . . . . . . . . . 12 ((𝑓:ω–1-1𝐴 ∧ (ω ∖ {∅}) ⊆ ω) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
2926, 27, 28sylancl 580 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅})))
30 f1oen3g 8178 . . . . . . . . . . 11 (((𝑓 ↾ (ω ∖ {∅})) ∈ V ∧ (𝑓 ↾ (ω ∖ {∅})):(ω ∖ {∅})–1-1-onto→(𝑓 “ (ω ∖ {∅}))) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
3125, 29, 30sylancr 581 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ω ∖ {∅}) ≈ (𝑓 “ (ω ∖ {∅})))
32 f1orn 6332 . . . . . . . . . . . . 13 (𝑓:ω–1-1-onto→ran 𝑓 ↔ (𝑓 Fn ω ∧ Fun 𝑓))
3332simprbi 490 . . . . . . . . . . . 12 (𝑓:ω–1-1-onto→ran 𝑓 → Fun 𝑓)
34 imadif 6153 . . . . . . . . . . . 12 (Fun 𝑓 → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
3516, 33, 343syl 18 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ (ω ∖ {∅})) = ((𝑓 “ ω) ∖ (𝑓 “ {∅})))
36 f1fn 6286 . . . . . . . . . . . . . 14 (𝑓:ω–1-1𝐴𝑓 Fn ω)
3736adantl 473 . . . . . . . . . . . . 13 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝑓 Fn ω)
38 fnima 6190 . . . . . . . . . . . . 13 (𝑓 Fn ω → (𝑓 “ ω) = ran 𝑓)
3937, 38syl 17 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ ω) = ran 𝑓)
40 fnsnfv 6449 . . . . . . . . . . . . . 14 ((𝑓 Fn ω ∧ ∅ ∈ ω) → {(𝑓‘∅)} = (𝑓 “ {∅}))
4137, 9, 40sylancl 580 . . . . . . . . . . . . 13 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → {(𝑓‘∅)} = (𝑓 “ {∅}))
4241eqcomd 2771 . . . . . . . . . . . 12 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ {∅}) = {(𝑓‘∅)})
4339, 42difeq12d 3893 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝑓 “ ω) ∖ (𝑓 “ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
4435, 43eqtrd 2799 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓 “ (ω ∖ {∅})) = (ran 𝑓 ∖ {(𝑓‘∅)}))
4531, 44breqtrd 4837 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
46 entr 8214 . . . . . . . . 9 ((ω ≈ (ω ∖ {∅}) ∧ (ω ∖ {∅}) ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
4724, 45, 46syl2anc 579 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
48 entr 8214 . . . . . . . 8 ((ran 𝑓 ≈ ω ∧ ω ≈ (ran 𝑓 ∖ {(𝑓‘∅)})) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
4919, 47, 48syl2anc 579 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}))
50 difexg 4971 . . . . . . . 8 (𝐴 ∈ V → (𝐴 ∖ ran 𝑓) ∈ V)
51 enrefg 8194 . . . . . . . 8 ((𝐴 ∖ ran 𝑓) ∈ V → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
525, 50, 513syl 18 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓))
53 disjdif 4202 . . . . . . . 8 (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅
5453a1i 11 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅)
55 difss 3901 . . . . . . . . . 10 (ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓
56 ssrin 3999 . . . . . . . . . 10 ((ran 𝑓 ∖ {(𝑓‘∅)}) ⊆ ran 𝑓 → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)))
5755, 56ax-mp 5 . . . . . . . . 9 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓))
58 sseq0 4139 . . . . . . . . 9 ((((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) ⊆ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) ∧ (ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
5957, 53, 58mp2an 683 . . . . . . . 8 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅
6059a1i 11 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)
61 unen 8249 . . . . . . 7 (((ran 𝑓 ≈ (ran 𝑓 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ ran 𝑓) ≈ (𝐴 ∖ ran 𝑓)) ∧ ((ran 𝑓 ∩ (𝐴 ∖ ran 𝑓)) = ∅ ∧ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∩ (𝐴 ∖ ran 𝑓)) = ∅)) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
6249, 52, 54, 60, 61syl22anc 867 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) ≈ ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)))
638frnd 6232 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ran 𝑓𝐴)
64 undif 4211 . . . . . . 7 (ran 𝑓𝐴 ↔ (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
6563, 64sylib 209 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓)) = 𝐴)
66 uncom 3921 . . . . . . 7 ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)}))
67 eldifn 3897 . . . . . . . . . . 11 ((𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓) → ¬ (𝑓‘∅) ∈ ran 𝑓)
68 fnfvelrn 6548 . . . . . . . . . . . 12 ((𝑓 Fn ω ∧ ∅ ∈ ω) → (𝑓‘∅) ∈ ran 𝑓)
6937, 9, 68sylancl 580 . . . . . . . . . . 11 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝑓‘∅) ∈ ran 𝑓)
7067, 69nsyl3 135 . . . . . . . . . 10 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
71 disjsn 4404 . . . . . . . . . 10 (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ ↔ ¬ (𝑓‘∅) ∈ (𝐴 ∖ ran 𝑓))
7270, 71sylibr 225 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅)
73 undif4 4197 . . . . . . . . 9 (((𝐴 ∖ ran 𝑓) ∩ {(𝑓‘∅)}) = ∅ → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
7472, 73syl 17 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}))
75 uncom 3921 . . . . . . . . . 10 ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = (ran 𝑓 ∪ (𝐴 ∖ ran 𝑓))
7675, 65syl5eq 2811 . . . . . . . . 9 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) = 𝐴)
7776difeq1d 3891 . . . . . . . 8 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (((𝐴 ∖ ran 𝑓) ∪ ran 𝑓) ∖ {(𝑓‘∅)}) = (𝐴 ∖ {(𝑓‘∅)}))
7874, 77eqtrd 2799 . . . . . . 7 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((𝐴 ∖ ran 𝑓) ∪ (ran 𝑓 ∖ {(𝑓‘∅)})) = (𝐴 ∖ {(𝑓‘∅)}))
7966, 78syl5eq 2811 . . . . . 6 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → ((ran 𝑓 ∖ {(𝑓‘∅)}) ∪ (𝐴 ∖ ran 𝑓)) = (𝐴 ∖ {(𝑓‘∅)}))
8062, 65, 793brtr3d 4842 . . . . 5 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → 𝐴 ≈ (𝐴 ∖ {(𝑓‘∅)}))
8180ensymd 8213 . . . 4 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴)
82 entr 8214 . . . 4 (((𝐴 ∖ {𝐵}) ≈ (𝐴 ∖ {(𝑓‘∅)}) ∧ (𝐴 ∖ {(𝑓‘∅)}) ≈ 𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
8313, 81, 82syl2anc 579 . . 3 (((ω ≼ 𝐴𝐵𝐴) ∧ 𝑓:ω–1-1𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
842, 83exlimddv 2030 . 2 ((ω ≼ 𝐴𝐵𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
85 difsn 4485 . . . 4 𝐵𝐴 → (𝐴 ∖ {𝐵}) = 𝐴)
8685adantl 473 . . 3 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → (𝐴 ∖ {𝐵}) = 𝐴)
87 enrefg 8194 . . . . 5 (𝐴 ∈ V → 𝐴𝐴)
884, 87syl 17 . . . 4 (ω ≼ 𝐴𝐴𝐴)
8988adantr 472 . . 3 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → 𝐴𝐴)
9086, 89eqbrtrd 4833 . 2 ((ω ≼ 𝐴 ∧ ¬ 𝐵𝐴) → (𝐴 ∖ {𝐵}) ≈ 𝐴)
9184, 90pm2.61dan 847 1 (ω ≼ 𝐴 → (𝐴 ∖ {𝐵}) ≈ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 384   = wceq 1652  wex 1874  wcel 2155  Vcvv 3350  cdif 3731  cun 3732  cin 3733  wss 3734  c0 4081  {csn 4336   class class class wbr 4811  ccnv 5278  ran crn 5280  cres 5281  cima 5282  Fun wfun 6064   Fn wfn 6065  wf 6066  1-1wf1 6067  1-1-ontowf1o 6069  cfv 6070  ωcom 7265  cen 8159  cdom 8160
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-ord 5913  df-on 5914  df-lim 5915  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-om 7266  df-1o 7766  df-er 7949  df-en 8163  df-dom 8164
This theorem is referenced by:  infdiffi  8772  infcda1  9270  infpss  9294
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